I did all of exactly what you are about to do a month ago on my partscaster (as well as previous other refins I have done).
I use a dremel without any attachments. Use both hands while sitting with your elbows on the table/bench and you'll have more control than you may think. It's small and light and the holes aren't going to be that deep or take very long to cut anyway. You should be able to keep it straight enough.

First I use the screw thread length to measure the depth I need to go on the drill bit and wrap a piece of masking tape around it.

I mark the hole position, put the bit in position and start the hole by turning the drill by hand first. You can even have the PG, Tuners and control plate in position when doing this. This puts a starting hole for the bit. Remove the pg, plate or tuner, put the drill bit in the little hole you just made, turn on the dremel and drill to the depth of the tape on the bit.

One trick I recommend is to countersink the hole slightly before putting any screws in. Since the hole is actually smaller than the oside diameter of the threads, when you screw it in the threads can/will crack and lift the finish off the guitar. In most cases the guard, plate or tuner will hide it but I have seen hugh flakes of finish come off a friends guitar that extended from under the PG. Just take a 60deg countersink or even a drill bit thats a bit larger than the thread diameter and, by hand, hold the shank of the bit in your left hand and spin the bit with your right (other way if your a lefty). Just shave off enough finish so the threads clear when the srew is put in. I usually go til I can just see the wood around the edge of the hole. Use a little wax or soap to lube the screw. It will go in a lot easier and you won't risk snapping of a screw head.

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